Two linear orderings are equimorphic if each can be embedded into the other. We prove that every hyperarithmetic linear ordering is equimorphic to a recursive one.
On the way to our main result we prove that a linear ordering has Hausdorff rank less than ω₁CK if and only if it is equimorphic to a recursive one. As a corollary of our proof we prove that, given a recursive ordinal α, the partial ordering of equimorphism types of linear orderings of Hausdorff rank at most α ordered by embeddablity is recursively presentable.
"Up to equimorphism, hyperarithmetic is recursive." J. Symbolic Logic 70 (2) 360 - 378, June 2005. https://doi.org/10.2178/jsl/1120224717